Finite Element Method applied to a monocomponent liquid adsorption model with non-linear isotherm

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1 GIM_00 Thrd Jo ofr of Iala Grou of omuaoal Mhas ad Ibro-a Amra Assoao of omuaoal Mhods Egrg F Elm Mhod ald o a mooomo lqud adsoro modl wh o-lar sohrm A. P. da S. VARGAS, M. B. HEKE, A. d P. SHEER,. H. MARHI, M. R. W. MAIE 3 Gradua Program o umral Mhods for Egrg, Fdral Uvrsy of Paraá, urba, Bral hmal Egrg Darm, Fdral Uvrsy of Paraá, urba, Bral 3 Fauly of hmal Egrg, Sa Uvrsy of amas, São Paulo, Bral ABSTRAT I hs arl h rsoluo ad umr rsuls of h lqud adsoro roblm fxd-bd ar show. Thy wr obad hrough wo umr modls, boh usg h F Elms Mhod (FEM) o sa dsrao ad for m doma o usd h ra-olso Mhod (FEM) ad h ohr usd h 4h Ordr Rug-Kua Mhod (RKFEM). Th mahmaal modl rad hs ar s u-dmsoal, omosd by aral dffral quaos ha dsrb h dffusv-ovv rasor. Th adsoro qulbrum s dsrbd by agmur s o-lar sohrm. Wh hs aroah was ossbl o sudy som of h uavodabl rrors from ay umr soluo. Th mhods mlmao was do wh MAPE VI sofwar, ad h rsuls obad wr qualavly aalysd ad omard wh xrmal daa.. ITRODUTIO Th alao of F Elms Mhods (FEM) o a gra vary of omlx roblms has b sudd by may ss a h rs. As xamls, h gohal [] ad vromal grg [7, 8, 9] suds of groud solus rasor, ad h gass adsoro alulao do by hmal grs [3]. I hs wor h FEM was ald o allow sa dsrao, osdrg a sf mooomo lqud adsoro modl as rfrrd h ars of Buso al [3] ad Shr []. For m dsrao was usd h ra-olso Mhod (FEM) or h 4h Ordr Rug-Kua Mhod (RKFEM). Adsoro s h ross whr a slv orao of o or mor omos of a gas as wll as lqud ar adsorbd o h surfa of a mro orous sold. Th mahmaal quao ha dsrbs h adsoro qulbrum s h agmur s quao, whr flud orao ad adsorba orao ar rlad [3]. Wh h r dvloms o umr omuao ad h d of mor ffvss smulao hqus, owrful sofwars of dyam hmal rosss smulao hav b dsgd maly o

2 dsllao ross rohmal dusry, som ohrs ors absoro ad xrao, bu a mora ross as h adsoro rmas sod la. Ths wor rooss h us of FEM o adsoro roblms. Th dvlod mahmaal modls wr mlmd wh MAPE VI sofwar, ad h rsuls obad wr qualavly aalysd ad omard wh xrmal daa. Th mahmaal modl, h FEM formulao ad h FEM ad RKFEM formulao ar show o so. So 3 shows h rsuls ad o so 4 h dsusso ad som fal omms ar rsd.. MATHEMATIA MODE Th dffuso o orous was h mahmaal modl usd for h adsoro, whr wo mass rasfr ross ar osdrd: - h xral mass rasfr from h bul lqud has o h sold surfa; - h ral dffuso sd h arl s orous. Th lqud adsoro roblm s dsrbd by mahmaal modl gv by h followg dffral quaos [3,]: ε v D ε K 3 R ( )( ε ) () 3 ε K s R ( ) ρ ( ε ) q () q aq m ( a ) (3) osdrg hs al ad boudary odos: F ( 0, ) (, ) 0 (,0) 0 (,0) 0 q (,0) 0 (4) whr s h orao h xral lqud has, s h orao of h lqud soluo h or of h adsorb arl, s m ad s h vral sa oorda. Equao (3) s gv by agmur s o-lar sohrm q aq m a (5)

3 whr q s h adsorba orao, q m s h maxmum adsorba orao h adsorb, ad a s a osa. For varaoal form, osdrg wghd rsduals, h quao () s wr as: Fd (,) suh as 0 h h α h β( ) h Dl 0 h 0 (6) whr h fuo h s dfd by h odos o (4) as h(0,)0 []. Th osas wr groud as follows: α 3 ε v / ε β K R ε (7).. umr Modls Th FEM has b udr valuao oghr wh som grao hqus usd o solv roblms o m doma. Som xamls a b foud o a vary of roblms volvg hyrbol ad/or arabol quaos wh suds ad dsussos abou aramrs varaos [4, 7, 8, 9,]. I h modl rsd, h FEM was ald o quao () ad wll b dmosrad h x sos. Equaos () ad (3) wr modfd o ada o h FEM ad RKFEM mhods, rsulg h m dd o-lar dffral quao: σ ( a) ( ) ( a ) Φ [ ] (8) od K σ 3 ε R ad Φ ρ aq s m ( ε ) ε (9).. Th F Elms Mhod ald o h adsoro olum Th aroxmad soluos for quao () s varaoal formulao, gv by quao (6), ar h bul h f dmsoal subsa of admssbl fuos, whh sasfy h homogous boudars odos ad ar smooh ough o wll df h grals a varaoal roblm, also allowg h frs drvavs o bom squar-grabl. A aro s rodud h (0,) rval rsulg sub-rvals, whr o F Elm volvg wo ods, ad, s bul o solv hs roblm by usg ws lar fuos for h rolao sd h lm. Th followg rolaos ar roosd for (,) oraos ad for h(,t) "vrual oraos": ( ), ( ) ( ) h(, ) ( ) H ( ) (0) whr - s h umbr of lms a h mom; 3

4 4 - (,) ar salar fuos obad from h soluo of a sysm of ordary dffral quaos () ad (4), wh uow fuos; -, H ad () ar bul has orao vor, "vrual oraos" ad osud rolao oraor omos, rsvly. Rlag hs rlaos o (6): l D β α 0 H β H () as qualy mus b sasfd for vry H omo, h: D β α } 0 β () for,...,. osdrg m D β α f β (3) quao () a b wr as 0 m f ara,.., (4) or 0 F K M (5) whr M, K, ad F ar marx ad vors formd by h omos m, ad f, wh valus varyg bw ad for ad for lm. For h fal osruo of h global marx, a boola marx rrsg h d rlao of lm was usd, as follows:

5 l T M K Y 0 (6) whr l s h umbr of lms. From (6) s obad h global sysm of dffral quaos: M K F 0 (7).3. Formulao usg FEM ad RKFEM mhods May grao hqus a b usd o solv m doma roblms. For h FEM alao o hs modl, xrsso (7) s assumd as: [ A]{ } [ B] (8) whr [ A ] M K [ B] M K { } [ F ] (9) I hs sudy, h marx ar alulad o for ah mom, obag (). Th s alulad wh FEM quaos (8) ad (9) for ah mom. Afr ha, h alulao follows a rav ross. For h RKFEM alao, h sam ross was usd, bu h marx wr alulad a ah RKFEM has. 3. RESUTS Th sudy of ad bd adsoro olums s basd o h aalys of m X bul has orao urvs, whh ar flud by olum gomry, oraoal odos ad qulbrum daa. Ths urvs, rfrrd as Brahrough urvs, rsul from h orao ouu moorg of a flud ha asss hrough a ad bd. Afr som m h bd boms saurad, ad h ouu orao aroahs h u orao. Th ara bhd h brahrough urv rrss h quay of adsorba rad h olum. Ths orrsods o a o o h qulbrum sohrm []. Th umr rsuls rsd hs so ar basd o h omarso wh Saasara xrmal rsuls [0] for h m-xyl omo. Th followg aramrs wr osdrd for hs sf roblm: T 9600s, 40m, D ρ.4e3, a 4. / mol, K.9E, ε.09m / s, q 0.4, ε m 0., R.75E 3, f 6.5E, v.98e, 78E. (0) Th rsuls grad by FEM ar show Fgur, as hos by RKFEM ar Fgur. Eah grah rrss h sa dsrao rfm for a sulad m rm. For h FEM, m rms of 00, 00, 50, 5 ad.5 sods wr sudd ad sa rfms wr obad for 0, 0, 40 ad 80 lms. For h RKFEM, m rms of 00, 50, 5,.5 ad 6.5 sods wr sudd ad sa rfms wr obad for 0, 0 ad 40 lms. 5

6 I a b od ha m rms of 00s ad 00s for FEM ad 00s for RKFEM ar hos ha br f wh xrmal daa. Aordg o Frgr ad Pér [5], sal ar mus b a wh rrg hs rsuls, osdrg ha dffr ds of msas a b hdd by rsuls ha aar o ovrg o xrmal daa. Alhough, wh msh rfm, h urvs d o dslam from xrmal daa, bu all of hm ovrg o h sam soluo, for boh mhods. Ths fa, howvr, a also da a modllg rror du o h hos aramrs. I a also b od ha h osllaos d o dsaar whl msh rfm s mrovd, wha mas ha hs mhod a b adusd ddg o h rfms, allowg furhr suds o h aramrs s. I h las grah of ah fgur s rsd h msh smulaous rfm, amg o d ossbl umr rrors. 4. DISUSSIO AD FIA OMMETS A frs sgh, h modl maulao was adqua o mhods alao, osdrg som smlfaos as h boudars odos. Th umr modl, sll udr sudy, showd good aorda wh xrmal daa, bu has som faors ha mus b br aalysd. I hs as, h os of Frgr ad Pér [5] hav b usd as a ool for rsuls aalyss, owg ha rrors ar uavodabl umr soluo. Wh mshs rfm s blvd ha rrors wr mmsd, as good ovrg was obad. Boh modls showd h sam bhavour, bu ryg o rah a aabl rror lvl for h roblm, h aalyss da ha FEM was mor arora ha RKFEM osdrg h osllaos rsd. Wh rfm s usd o boh varabls, h rsuls ar fr from osllaos ad hr s o d o us ohrs arfas ommoly ulsd hs d of aalyss. Th urv dslam, omarg wh xrmal daa, mus b br sudd. Th FEM shows gra advaags from h omuaoal o of vw, allowg h us of mor rfd F Elms mshs, ad has, also, ough odos o suor suds o bary or mulomos adsoro. 6

7 a) b),,0 DETA 00, DETA 00,0 / 0 0, 0 Elm 0 Elm 40 Elm 80 Elm Exr. / 0 0, 0 Elm 0 Elm 40 Elm 80 Elm Exr (s) (s) ) d), DETA 50, DETA 5,0,0 / 0 0, 0 Elm. 0 Elm. 40 Elm. 80 Elm. Exrm. / 0 0, 0 Elm 0 Elm 40 Elm Exrm (s) (s) d) ), DETA.5, SIMUTAEOUS,0,0 / 0 0, 0 Elm 0 Elm 40 Elm Exrm. / 0 0, D50/El0 D5/El0 D,5/El40 Exrm (s) (s) 7

8 Fgur : Brahrough urv smulad by FEM for a) Ä 00s, b) Ä 00s, ) Ä 50s, d) Ä 5s, ) Ä.5s ad f) Smulaous rfm (sa ad m). a) b),,0 DETA 00,,0 DETA 50 / 0 0, -0, Dla 00 Exrm. / 0 0, Elm 0 Elm 0 Exrm (s) (s) ) d),,0 DETA 5,,0 DETA.5 / 0 0, Elm 0 Elm 0 Exrm. / 0 0, Elm0 Elm0 Exrm (s) (s) ) f),4,,0 DETA 6.5,,0 SIMUTAEOUS / 0 0, Elm 0 Elm 0 Elm 40 Exrm. / 0 0, D5/E0 D,5/E0 D6,5/E40 Exrm (s) (s) 8

9 Fgur : Brarhough urv smulad by RKFEM for a) Ä 00s, b) Ä 50s, ) Ä 5s, d) Ä.5s, ) Ä 6.5s ad f) Smulaous rfm (sa ad m). REFEREES [] G A.K.., Small J..: Us of ould F Elm aalyss saurd sol roblms, J. umr. Aal. Mh. Gomh I., Vol. 4,.73-94, 000. [] Br E.B, ary G.F, Od J.T., F Elms - A Iroduo; Pr-Hall,.J., USA, 98. [3] Buso A., Paralla A., Troa A..: Th soluo of h dyam adsoro bds usg h F Elm Mhods; omu. hm. Egg, Vol.,. 47-5, :, 988. [4] Frrra W. R., Fors M., Juho R. A.., osa. M. S.: oama dsrso roblms saurd sols: soluos by dffusoal f lms; s Ibra a Amra ogrss o omuaoal Mhods Egrg, Ro d Jaro, Brasl, 00, Absras,.6. [5] Frgr J. H., Pr M.: omuaoal Mhods for Flud Dams; Srgr, Brlm, 999. [6] Flayso B.A., o-ar Aalyss hmal Egrg, MGraw- Hll, w Yor, 980. [7] Hossa M.A., Mah A.S., ra-olso-galr modl for rasor groudwar: Rfd rra for auray, Al. Mah. omu., Vol. 05,.73, 999. [8] Hossa M.A., Taha M.R.: Smulag advv-dsrsv rasor by F Elms rra for auray of a xl Rug-Kua Mhod, Al. Mah. omu., Vol., , 000. [9] Hossa M.A., Yog D. R.: Modlg oama rasor groudwar: A omd f lm mhod, Al. Mah. omu., Vol. 96, , 998. [0] Saasara E., Morbdll M., Srvda A., Sor G., arra S.: Sarao of xyls o y ols.. Brahrough urvs ad hr rrao, Id. Eg. hm. Pross. Ds., Vol., , 98. [] Shr A. P.: Dsvolvmo d um ssma ara smulação omação do rosso d adsorção ara avalação da saração d msuras líqudas, Ts (Douorado m Eghara Químa), amas, UIAMP, 00., 00. [] Sgh K.: Grd adusm for arabol sysms basd o a osror rror smas, J. om. ad Al. Mah., Vol. 63, , 995. [3] Ruhv D.M., Prls of Adsoo ad Adsoro Prosss, Joh Wly, w Yor,